In this paper we prove that it is possible to use techniques specific to electromagnetic field synthesis in the study of some electrical circuits. The definition of a circulant network will be presented. The system matrix of such a network is a circular matrix, which allows an analytical evaluation of all the eigenvalues and all the singular values. Resonance frequencies can then be calculated exactly as will be demonstrated on passive and active circulant networks

De Mey, Gilbert

Micu, Dan

English

Ghent University Academic Bibliography

Électrotechnique et électroénergétique THE CONDITION NUMBER FOR CIRCULANT NETWORKS DAN MICU1, GILBERT DE MEY2 Key words: Circulant networks, Eigenvalues, Singular values, Condition number. In this paper we prove that it is possible to use techniques specific to electromagnetic field synthesis in the study of some electrical circuits. The definition of a circulant network will be presented. The system matrix of such a network is a circular matrix, which allows an analytical evaluation of all the eigenvalues and all the singular values. Resonance frequencies can then be calculated exactly as will be demonstrated on passive and active circulant networks. 1. INTRODUCTION The condition number is an essential tool in the field synthesis problem [1]. Using conditioning also in the study of electrical circuits [2] could prove beneficial. A condition number close to unity guarantees the system matrix is well conditioned. High values of condition number indicates ill conditioning which can be either due to numerical problems but also due to the fact the electric network is not stable. Examples for this are the circulant networks. A circulant network is defined as a series connection of identical basic cells connected in a closed loop. It is known [3] that a closed loop can give rise to resonance or instability especially if active elements are involved.  As it will be shown further, the system matrix of such networks turns out to be a circular matrix [4], the reason the name circulant networks has been chosen. The major property is that all the eigenvalues of a circular matrix can be calculated analytically. This offers the unique possibility to evaluate condition numbers of the system matrix exactly, so that eventual ill conditioning or resonances can be evaluated exactly. The importance of circulant networks derives from the fact that if in a chain of identical quadrupoles the matrix of the fundamental parameters at the n-th power is the identity matrix then a chain of n such quadrupoles can be studied as a                            1 Technical University of Cluj-Napoca, d_o_micu@yahoo.com 2 University of Ghent, Department of Electronics, Ghent, Belgium Rev. Roum. Sci. Techn. – Électrotechn. et  Énerg.,  58, 2, p. 115–122, Bucarest, 2013 116 Dan Micu, Gilbert De Mey 2   circulant network. The applications can be extended to the study of filters or transmition lines. 2. BASIC THEORY Let us consider the circulant network shown in Fig. 1. The unit cell is composed of an impedance Z and an admittance Y. The current sources Ii can be diferent. A circulant network implies that the chain of Fig. 1 must form a closed loop or the node n + 1 = node 1. The electromagnetic field quasi-stationary equations lead to the following equation:                   iiiiii IVYZVVZVV =+−−− +− 11 , (1) for i = 1, 2, ... n. If i = 1 then Vi-1 has to be replaced by Vn and if i = n then Vi+1 has to be replaced by V1 in (1).  Z Y Ii Vi Vi+1 i i+1  Fig. 1 – The unit cell of circulant network. (1) can be rewritten as:                   (2+Z Y)Vi –Vi-1 –Vi+1 = ZIi. (2) Or in matrix form [5, 6]:                   [A][V] = [Z][I]. (3) The system matrix [A] is then: +−−−+−−+−−+−−+−−−+YZYZYZYZYZYZ211121............1......21121121112, (4) 3 The condition number for circulant networks 117   [A] is obviously a circular n x n matrix, the eigenvalues of which can be calculated analytically. 3. EIGENVALUES ANALYSIS The eigenvalues of a circular matrix [A] can be evaluated as follows:                ∑ = ==λ nk k ZYA1 ,10 , (5)         1 21, 10πλ exp 2πj 4sinnm kkmk mA ZYn n−+= = = +  ∑ . (6) If Z Y is real and positive and n even the maximum eigenvalue occurs for m = n/2 or λn/2 = Z Y + 4. The condition number is then (for circular matrix):                        κ = |λn/2|/|λ0| = |Z Y+4|/|Z Y|, (7) or the largest eigenvalue (in absolute value) divided by the smallest one. Generally the condition number is defined with the so-called singular values σi. These are defined by [2]:                   ,ii µ=σ  (8) where µi are the eigenvalues of the matrix [A]H[A], [A]H being the Hermitean (the transpose conjugate) [4] of the system matrix [A]. If  1210 ... −σ≤≤σ≤σ≤σ n  the condition number κ is then defined by:                  01σσ= −nk . (9) High values of κ indicates ill conditioning which can be either due to numerical problems but also due to the fact the electric network is not stable (oscillator e.g.).  An important difference between eigenvalues and singular values is that the latter ones are always real and positive numbers. [A]H[A] is a circular matrix too and all the elements are real numbers. The eigenvalues µi of [A]H[A] are found similarly by using (5) or (6):         **1 ,10YZYZAAnk kH == ∑ =µ  (10)          11, 10µ exp 2πjn Hm kkmkA An−+= = =  ∑  (11) 118 Dan Micu, Gilbert De Mey 4          ( )nmnmYZYZYZYZ π+π+= 22**** sin16sin4 =  π+π+=nmYZnmYZ 2**2 sin)sin4( . It can be easily verified that               2mm λ=µ . (12) We note Z = r+jx, Y = g+jy so:            Z Y = rg–xy+j (ry+gx) = a+j b. (13) Let p and q corresponding to:  222max sin4 bnpap + π+=µ=µ ; 222min sin4 bnqaq + π+=µ=µ .   The condition number is:          qpK µµ= .  (14) If a > 0 the condition number indicates a well conditioned system matrix. In this case 2np =  and q = 0 (the (7) formula is found once again for even n and b = 0. The ill-conditioned system is obtained only [7] for b = 0 and a ∈ [– 4, 0] so  Z  =  jωL and Y  = jωC, p = 0 and 0sin4 2 =π+nqa  so the resonance condition is:               nqLCπ=ω 22 sin4 . (15) For example if n = 3 the condition is:                ω2 LC = 3 . (16) After the triangle-star transformation in the circulant network of Fig. 1, the circuit is presented in Fig. 2, where ii ICE ω= j1 . With (16) we have serie resonance in Fig. 2 and if the sources are different the network is in great difficulty. 5 The condition number for circulant networks 119    C Ei CE3CE2 Fig. 2 – Resonance in circulant network with 3 cells. For q = 0 one gets then a resonance at "DC" which means physically that all the capacitors in the lumped LC are being charged to the same constant but arbitrary voltage. The singular values are now:                 LCo2ω=σ ,                   nmLCmπ−ω=σ 22 sin4| . (17) 4. APPLICATION TO AN ACTIVE NETWORK  Let us consider now the circulant network shown in Fig. 3. Each cell consist of an operational amplifier [3] and two impedances Z1 and Z2. The operational amplifier is considerer as an ideal one: infinite gain, infinite input resistance and zero output resistance.  Vi Vi+1 i i+1 Z1 Z2 _+ Fig. 3 – The unit cell of active network. As can be found in several textbooks one has:              α−=−=+121ZZVVii ,  (18) 120 Dan Micu, Gilbert De Mey 6   or            α Vi +Vi+1 = 0. (19) The system matrix [A] is now: [A] = αααααα11..................111.  (20) Obviously, (20) is a circular matrix, the eigenvalues [4] of which are easily found be using (5) and (6):              10 +α=λ , (21)                   π+α=λnmm j2exp . (22) Example 1. As a first practical example we consider a circulant network composed of inverters Z1 = Z2 = R or α = 1. We have then:           20 =λ , (23)                π+=λnmjm 2exp1 , (24)                 nmnmmπ= π+=λ cos22cos12 . (25) λm  can be zero for n even and m = n/2. The network has then an infinite number of solutions:               Vi  = V0  (i even) and Vi = –V0  (i odd), (26) where V0 can be any value. If n is odd, the only possible solution for the network is Vi  = 0 for all values of i. For n odd the smallest eigenvalue is evaluated as:                nn 2sin2cos12minπ= π−=λ . (27) The condition number [2] is then: 7 The condition number for circulant networks 121                 nk2sin1min0π=λλ= , (28) which can be quite hight for large values of the number of cells n. Example 2. A second example involves an RC network with Z1 = R and  Z2 = 1/jωC. This gives rise to α = 1/jωRC. The eigenvalue are then:            222011j11CRRC ω+=ω+=λ , (30)            nmRCCRmπω−ω+=λ2sin211 222 .  λm can be zero if ωRC = 1 and sin(2πm/n) = 1 which is only possible if n is a multiple of 4. The circuit is then oscillating [3] at an angular frequency 1/RC. Example 3. As a third example we consider a parallel connection of a resistor R and a capacitor C for Z2 and a single resistor for Z1:               21RZ = , (31)               RCRZ ω+= j12 . (32) We have now:              RCω+=α j12. (33) The eigenvalues are then found to be:                ( ) 112122220+ω++=λCR, (34)    nmCRRCnmCRCRmπω+ω−πω+++ω+=λ2sin1222cos122112222222222. (35) One can easily verify that for ωRC = 1 the eigenvalue λm = 0 on condition 2πm/n = 2π 3/8 or:                     83=nm. (36) 122 Dan Micu, Gilbert De Mey 8   If the circulant network contains 8 cells, the eigenvalue, 03 =λ  and the network will be oscillating at a frequency ω = 1/RC. The physical explanation is quite obvious. If ωRC = 1 each cell has a gain equal to unity and a phase shift of exactly  2π/8. Hence the 8 cell will provide a total phase shift 2π so that oscillation can start. The same phenomenon will occur if n is any multiple of 8. 5. CONCLUSIONS It has been proved in this paper that the system matrix of a so called circulant network turns out to be a circular matrix. These matrices have the major advantage that all the eigenvalues can be calculated analytically. It becomes then possible to analyse phenomena like resonance exactly, without any disturbance due to ill conditioned matrices. It was also observed that the results obtained from the eigenvalues or from the singular values are all in perfect agreement. This is a consequence of the fact that the singular values are equal to the absolute values of the eigenvalues for the circular matrix. The results and methods used can be applied in the study of quadrupoles, filters and transmition lines. Received on December 12, 2012 REFERENCES 1. D. D. Micu, D. Micu, A. Ceclan, E. Simion, Synthesis Method of an Inductive Sensor using Tikhonov Regularization Procedure, International Journal of Applied Electromagnetics and Mechanics, IJAEM, 26, pp. 243–251, 2007. 2.  D. Micu, Lazăr Chira, Elemente de teoria circuitelor electrice, Edit. Dacia, Cluj-Napoca, 2010. 3. G. De Mey, Potential calculations in Hall plates, Advances in Electronics and Electron Physics, 61, pp. 1–61, 1983. 4.  R.A. Horn, C. R. Johnson, Analiza matricială, Edit. Theta, Bucureşti, 2001. 5. C. Flueraşu, Corina Flueraşu, Circuite electrice. Teorie, modelare, simulare, Edit. Printech, Bucuresti, 2008. 6. F. I. Hanţilă, M. Maricaru, R. M. Ciuceanu, L. Corlan, Harmonic analysis of circuits with nonlinear resistive elements, Rev. Roum. Sci. Techn. – Électrotechn. et  Énerg., 4, pp. 377–340, 2012. 7. C. Marin, F. Constantinescu, Fault diagnosis in analog circuits using approximate symbolic pole/zero expressions, Rev. Roum. Sci. Techn. – Électrotechn. et  Énerg., 1, pp. 20–29, 2012. 

The condition number for circulant networks

https://biblio.ugent.be/publication/5851631/file/5851637.pdf

The condition number for circulant networks

Abstract

Similar works

Full text

Available Versions

Ghent University Academic Bibliography